Chaos Theory and Computers

Date: 2/4/96 at 17:35:0
From: Anonymous
Subject: Chaos Theory

I am not really trying to get the answer to a question, I am merely 
trying to make sense of a formula.  

I go to Cunningham Middle School in Presque Isle, Maine, and have 
recently been studying chaos theory. I have happened upon an equation 
that can create a fractal that demonstrates the theory graphically.  Now, 
since the computer cannot learn, and chaos theory is based on a lack of 
pattern, how is the computer able to compute this?  

The equation contains the x and y coordinates (making it 2D), but this is 
something I am unable to comprehend - i.  Isn't the variable i an imaginary 
number?  How is the computer able to use its pattern-based Operating 
System and FPUs to generate the algorithms associated with the variable i?  
It seems as if either the fractal (or graph, really) is off base, or the entire 
OS is off base.  Any ideas?

Date: 2/5/96 at 9:53:22
From: Doctor Ethan
Subject: Re: Chaos Theory

Hello,

Well, I can try to help you out.  First, let's deal with the imaginary 
number issue.

In this case i is not a variable.  i is the notation for the 
Square root of -1 so i is defined by

i * i = -1.

So, although the computer probably cannot inherently do complex 
arithmetic, it would be easy to design a program that could.

Now on to your other questions.

You are right, the computer doesn't learn anything, and to a degree you 
are right that Chaos lacks pattern.

However, if you have a formula that is chaotic (and some pretty simple 
ones are) then the computer doesn't need to do anything but follow the 
formula.  It is the formula rather than the computer that is chaotic. I 
can't say much about the actual formula since I don't know what it 
is; however, many chaos patterns are generated by recursion, which means 
doing the same process over and over.  Computers are very good at this, 
and they allow us to do the same things over and over thousands of 
times. This means that we can follow these patterns to see if they 
become chaotic.

I hope this has helped a little.  If you want more help, write back and 
maybe send us the formula that you have been using.

-Doctor Ethan,  The Math Forum